I'm using it to mean "arg(z) is a rational multiple of 2 pi", or equivalently "some power of z is real" or "z / abs(z) is a root of unity".
Sent: Wednesday, August 29, 2018 at 8:47 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] integers that are either square or thrice a square, where do such beasts hide?
With "rational arguments" — not sure I know what that means.
Eisenstein integers should be the ring Z[w] where w = -1/2 + i*sqrt(3/4).
Since Z[w] = {K + L*w | K, L in Z} we can compute
|K + L*w|^2 = (K - L/2)^2 + (3/4)*L^2
= K^2 - KL + L^2
(which is what I remembered).
I'm still in the dark over what rational arguments are, and how K^2 - KL + L^2 squares with what Adam wrote.
—Dan
Adam Goucher wrote: ----- They're precisely the norms of Eisenstein integers with rational arguments.
Wouter Meeussen wrote: -----
do they occur anywhere naturally? in combinatorics or in number theory? I bumped into them by accident looking at bracelet-stuff.
-----
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